Convergence of multilocus systems under weak epistasis or weak selection

Journal of Mathematical Biology

Volumn 38, Issue 2, 1999, Pages 103-133

  Nagylaki, Thomas ^a   Hofbauer, Josef ^b   Brunovský, Pavol ^c  

^a UNIVERSITY OF CHICAGO   (United States)

^b UNIVERSITY OF VIENNA   (Austria)

^c Physics and Informatics   (Slovakia)

Author keywords

Chain recurrence; Convergence; Epistasis; Invariant manifold; Quasi linkage equilibrium; Recombination; Selection

Indexed keywords

ALLELE; ARTICLE; BIOLOGICAL MODEL; DOMINANT GENE; GENETIC EPISTASIS; GENETIC LINKAGE; GENETIC SELECTION; GENETICS; MOLECULAR EVOLUTION; MULTIGENE FAMILY;

ALLELES; EPISTASIS, GENETIC; EVOLUTION, MOLECULAR; GENES, DOMINANT; LINKAGE (GENETICS); MODELS, GENETIC; MULTIGENE FAMILY; SELECTION (GENETICS);

EID: 0033071562     PISSN: 03036812     EISSN: None     Source Type: Journal    
DOI: 10.1007/s002850050143     Document Type: Article

References (45)

1. Akin, E.: The Geometry of Population Genetics. (Lecture Notes in Biomathematics, vol. 31) Berlin: Springer 1979
2. Akin, E.: Cycling in simple genetic systems. J. Math. Biol. 13, 305-324 (1982)
3. Akin, E.: Hopf bifurcation in the two-locus genetic model. Mem. Amer. Math. Soc., vol. 44, No. 284, 1983
4. Akin, E.: Cycling in simple genetic systems: II. The symmetric cases. In: Kurzhansky, A., Sigmund, K. (eds.) Dynamical Systems. (Lecture Notes Economics and Mathematical Systems, vol. 287, pp. 139-153) Berlin: Springer 1987
5. Akin, E.: The General Topology of Dynamical Systems. Providence, R.I.: Amer. Math. Soc. 1993
6. Barton, N. H., Turelli, M.: Adaptive landscapes, genetic distance and the evolution of quantitative characters. Genet. Res. 49, 157-173 (1987)
7. Baum, L. E., Eagon, J. A.: An inequality with applications to statistical estimation for probabilistic functions of a Markov process and to a model in ecology. Bull. Amer. Math. Soc. 73, 360-363 (1967)
8. Bodmer, W. F., Felsenstein, J.: Linkage and selection: theoretical analysis of the deterministic two locus random mating model. Genetics 57, 237-265 (1967)
9. Brunovský, P., Meravý, P.: Solving systems of polynomial equations by bounded and real homotopy. Numer. Math. 43, 397-418 (1984)
10. Chow, S. N., Hale, J.: Methods of Bifurcation Theory. Berlin: Springer 1982
11. Conley, C.: Isolated invariant sets and the Morse index. NSF CBMS Lecture Notes 38. Providence, R. I.: Amer. Math. Soc. 1978
12. Ewens, W. J.: A generalized fundamental theorem of natural selection. Genetics 63, 531-537 (1969a)
13. Ewens, W. J.: Mean fitness increases when fitnesses are additive. Nature 221, 1076 (1969b)
14. Ewens, W. J.: Remarks on the evolutionary effect of natural selection. Genetics 83, 601-607 (1976)
15. Ewens, W. J., Thomson, G.: Properties of equilibria in multi-locus genetic systems. Genetics 87, 807-819 (1977)
16. Fenichel, N.: Persistence and smoothness of invariant manifolds for flows. Ind. Univ. Math. J. 21, 193-226 (1971)
17. Garay, B. M.: Discretization and some qualitative properties of ordinary differential equations about equilibria. Acta Math. Univ. Comenianae 62, 249-275 (1993)
18. Garay, B. M.: On structural stability of ordinary differential equations with respect to discretization methods. Numer. Math. 72, 449-480 (1996)
19. Garay, B. M., Hofbauer, J.: Chain recurrence and discretization. Bull. Austral. Math. Soc. 55, 63-71 (1997)
20. Hastings, A.: Stable cycling in discrete-time genetic models. Proc. Natl. Acad. Sci. USA 78, 7224-7225 (1981)
21. Hastings, A.: Multilocus population genetics with weak epistasis. I. Equilibrium properties of two-locus two-allele models. Genetics 109, 799-812 (1985)
22. Hastings, A.: Multilocus population genetics with weak epistasis. II. Equilibrium properties of multilocus models: what is the unit of selection? Genetics 112, 157-171 (1986)
23. Hirsch, M. W., Pugh, C., Shub, M.: Invariant Manifolds. (Lecture Notes in Mathematics, vol. 583) Berlin: Springer 1977
24. Hofbauer, J.: Gradients versus cycling in genetic selection models. In: Aubin, J. P., Saari, D., Sigmund, K. (eds.) Dynamics of Macrosystems. (Lecture Notes in Economics and Mathematical Systems, vol. 257, pp. 90-101) Berlin: Springer 1985
25. Hofbauer, J., Iooss, G.: A Hopf bifurcation theorem for difference equations approximating a differential equation. Monatsh. Math. 98, 99-113 (1984)
26. Hughes, P.J., Seneta, E.: Selection equilibria in a multiallelic single-locus setting. Heredity 35, 185-194 (1975)
27. Karlin, S., Liberman U.: The two-locus multi-allele additive viability model. J. Math. Biol. 5, 201-211 (1978)
28. Karlin, S., Liberman, U.: Representation of nonepistatic selection models and analysis of multilocus Hardy-Weinberg equilibrium configurations. J. Math. Biol. 7, 353-374 (1979)
29. Kimura, M.: The attainment of quasi-linkage equilibrium when gene frequencies are changing by natural selection. Genetics 52, 875-890 (1965)
30. Kingman, J. F. C.: A mathematical problem in population genetics. Proc. Camb. Phil. Soc. 57, 574-582 (1961)
31. Kun, L. A., Lyubich, Yu. I.: Convergence to equilibrium under the action of additive selection in a multilocus multiallelic population. Sov. Math. Dokl. 20, 1380-1382 (1979)
32. Kun, L. A., Lyubich, Yu. I.: Convergence to equilibrium in a polylocus polyallele population with additive selection. Probl. Inf. Trans. 16, 152-161 (1980)
33. Lyubich, Yu. I.: Basic concepts and theorems of evolutionary genetics of free populations. Russ. Math. Surv. 26, 51-123 (1971)
34. Lyubich, Yu. I.: Mathematical Structures in Population Genetics. (Biomathematics, vol. 22) Berlin: Springer 1992
35. Moran, P. A. P.: On the nonexistence of adaptive topographies. Ann. Hum. Genetics 27, 383-393 (1964)
36. Nagylaki, T.: The evolution of one- and two-locus systems. II. Genetics 85, 347-354 (1977)
37. Nagylaki, T.: Rate of evolution of a character without epistasis. Proc. Natl. Acad. Sci. USA 86, 1910-1913 (1989a)
38. Nagylaki, T.: The maintenance of genetic variability in two-locus models of stabilizing selection. Genetics 122, 235-248 (1989b)
39. Nagylaki, T.: Introduction to Theoretical Population Genetics. (Biomathematics, vol. 21). Berlin: Springer 1992
40. Nagylaki, T.: The evolution of multilocus systems under weak selection. Genetics 134, 627-647 (1993)
41. Passekov, V.P.: Asymptotic analysis of selection in a multilocus, multiallelic population. (In Russian) Dokl. Akad. Nauk SSSR 277, 1338-1341 (1984)
42. Shahshahani, S.: A new mathematical framework for the study of linkage and selection. Mem. Amer. Math. Soc. Vol. 17, No. 211, 1979
43. Wright, S.: The analysis of variance and the correlations between relatives with respect to deviations from an optimum. J. Genet. 30, 243-256 (1935)
44. Wright, S.: The distribution of gene frequencies in populations. Proc. Natl. Acad. Sci. USA 23, 307-320 (1937)
45. Wright, S.: Evolution and the Genetics of Populations, Vol. 2. The Theory of Gene Frequencies. Chicago: The University of Chicago Press 1969